14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 1

Single Particle Motion

OverviewElectromagnetic fields,Lorentz-force, gyration and guiding center,drifts,adiabatic invariants.

Pre-requisites:Energy density of the particle population smaller than energy density of the field,All spatial and temporal changes in the fields are small/slow compared to the characteristic scales of particle motion.

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 2

Application: Radiation Belts

http://www.dartmouth.edu/~physics/cism/science/dartmouthmodel.html

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 3

Single Particle Motion: Overview

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 4

Electromagnetic Fields 1:Pre-requisite: vacuum, ε=µ=1.Maxwell´s laws:

Gauß´ law of the electric field

Gauß´ law of the magnetic field

Faraday´s law

Ampere´s law

Magnetic flux

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 5

Electromagnetic Fields 2:

Transition differential to integral form:

Gauß´ theorem:

Stokes´ theorem:

Transformation of electromagnetic fields:

general:

Non-relativistic:

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 6

Electromagnetic Fields 3:

Generalized form of Ohm´s law:

Energy density in the electromagnetic field:multiply Faraday by B and integrate over V:

rewrite (energy density now on the left hand side):

Poynting vector (energy flux density)

In matter:

Energy density electromagnetic field

>0: Ohmic losses, <0: sources

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 7

Lorentz Force

Lorentz force (general form, all electromagnetic fields)

Vanishing electric field (first integral of motion; elementary, only numerical exercise)

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 8

Gyration 1:

Lorentz force (equation of motion):Equation of motion (components):

Cyclotron frequency:

Larmor radius:

Example: T=1keV, B=1T; electrons: v=1.87E6 km, r=0.1 mm,ω=1.8E11/s; protons: v=4.37E5 km, r=4.6 mm, ω=1E8/s

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 9

Useful Definitions:

Magnetic rigidity:

Pitch angle:

Magnetic moment:

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 10

Gyration 2:

Relativistic quantities:

Example: proton with 1E20 eV, B = 3E-10 T, r = 1E21 mLocal gyro radius:

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 11

Drift in Electromagnetic Fields

Guiding center:

General drift:

Electric field perpendicular to magnetic field:

Gravitational plus magnetic field:

Gradient drift:

Curvature drift:

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 12

General Form of Drift

Idea: transformation into a frame of reference in which motion is reduced to:

Arithmetics (F⊥B):

Result: gyration in the new frame of reference⇒ general equation for the drift velocity:

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 13

Drifts Summarized

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 14

Drift with Changes in Energy

Drift leads particles to a different potential ⇒ acceleration

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 15

Adiabatic Invariants

Pre-requisite: spatial and temporal changes in the fields small compared to the corresponding scales of particle motion:

Gyration (time):

Field-parallel motion:

Gyro orbit (radius):

Magnetic moment:

Longitudinal invariant:

Flux invariant:

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 16

Magnetic Mirrors

Basics: the magnetic moment is constantRestoring force in the mirror point because magnetic field line not perpendicular to the plane of gyration

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 17

Magnetic Bottle

Two magnetic mirrors combined:Oscillation of the guiding center between the mirror pointsApplication: radiation belts (particles trapped inside the magnetosphere)Application longitudinal invariant: 1st order Fermi acceleration

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 18

Application: Radiation Belts

1. Magnetic moment µ2. Longitudinal invariant3. Flux invariant

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 19

Time Scales: Magnetosphere

Gyration: kHzLongitudinal oscillation: secondsDrift: 15 min (1000 s)

14/04/2005 Space Physics SS 2005 - Chap. 2: Single Particle Motion 20

Dynamics of the Radiation Belts

L-Shell: distance field line to center of Earth in equatorial planeIntensity on fixed field line (L-shell) shows strong temporal variationsIn particular rather abrupt depletion followed by a slower recovery of the particle population

http://lasp.colorado.edu/stp/research/research_main.html